The convergence of Euler products
type: Item
instance of: scholarly article
title: The convergence of Euler products
zbMATH Open document ID: 0768.46009
DOI: 10.1016/0022-1236(92)90103-P
published in: Journal of Functional Analysis
publication date: 1993-01-16T00:00:00Z
review text: The author's point of departure is Riemann's hypothesis that the classical zeta-function \(\zeta(s)\) has no zeros in the half-plane \(\text{Re }s>1/2\). He says: `The present work supports the conjecture that the Dirichlet zeta-function \(\zeta_ \chi(s)\) has no zeros in the half-plane \(\text{Re }s>1/2\) when \(\chi\) is a non-principal character modulo \(r\). The results do not apply without modification to the principal character.' His method is to associate certain Hilbert spaces of entire functions with \(\zeta_ \chi(s)\) using an interpretation of the Euler product for \(\zeta_ \chi(s)\). Some of these Hilbert spaces, which he calls Sonine spaces, rely upon properties of the Hankel transformation. The author obtains recurrence relations, differential equations, and asymptotic behaviour for the defining functions, but the results are too technical to state here. Modified versions of his results are derived for the classical zeta-function \(\zeta(s)\) with principal character modulo 1. They extend his earlier results such as [Bull. Am. Math. Soc., New Ser. 15, 1-17 (1986; Zbl 0602.10028)].
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 90039
zbMATH Keywords: zeta-function
zbMATH Keywords: Hilbert spaces of entire functions
zbMATH Keywords: Euler product
zbMATH Keywords: Sonine spaces
zbMATH Keywords: Hankel transformation
zbMATH Keywords: recurrence relations
zbMATH Keywords: differential equations
zbMATH Keywords: asymptotic behaviour
Wikidata QID: Q56566809
reviewed by: James V. Whittaker
MaRDI profile type: Publication
cites work: Self-reciprocal functions
cites work: The Riemann hypothesis for Hilbert spaces of entire functions
cites work: Complementation in Krein Spaces
cites work: Sonine spaces of entire functions
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description: scientific article; zbMATH DE number 90039